Characterization of MCD-spaces by uniform distance-shell sizes

Prove or disprove that a finite ultrametric space with center of distances \(C(X)=\{c_0,\ldots,c_n\}\), where \(0=c_0<\cdots<c_n\), is an MCD-space of cardinality \(2^n\) if and only if every point has exactly \(b_0=1\) points at distance \(c_0\) and \(b_i=2^{i-1}\) points at distance \(c_i\) for \(1\leq i\leq n\).

Background

This conjecture proposes a precise local characterization of the global extremal condition defining MCD-spaces. The prescribed shell sizes double across successive nonzero center-of-distances values, yielding total cardinality 2n2^n.

The statement is presented as a conjecture and is not established in the paper.

References

Then following statements are equivalent.

— Maximal Center of Distances of Finite Ultrametric Spaces and Perfect Binary Trees  (2609.11384 - Dovgoshey et al., 10 Sep 2026) in Section 5, Conjecture \ref{tcedg}

The authors believe that the following two conjectures are true. Let $X$ be a finite set. Then the following statements are equivalent.

— On the Graph Corresponding to Center of Distances of Metric Space  (2610.11596 - Dovgoshey et al., 8 Oct 2026) in Conjecture 1, Section 4 (Expected results)